On the Eigenvalues of the $p\&q-$ Fractional Laplacian

Abstract

We consider the eigenvalue problem for the fractional $p\&q-$Laplacian \begin{equation} \left\{\begin{aligned} (- Δ)_p^{s}\, u + μ(- Δ)_q^{s}\, u+ |u|^{p-2}u+μ|u|^{q-2}u=λ V(x)|u|^{p-2}u\quad & \text{in } Ω\\ u=0\quad& \text{in}\quad\R^N\backslashΩ, \end{aligned}\right. \end{equation} where $\Omega$ is an open bounded, and possibly disconnected domain, $\lambda \in \mathbb{R}$, $1<q<p<\frac{N}{s}$, $\mu >0$ with a weight function in $L^{\infty }(\Omega )$ that is allowed no change sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter $\mu \to 0^{+}.$ In addition, a stability property of eigenvalues as $s\to 1^{-}$ is established.